Rake, splay and sighting the result
What do a carpenter, who is trying to determine the length and cut angles of a hip rafter and a chair maker who is laying out the rake and splay of a plank seat chair for boring, have in common? As it turns out, quite a lot.
Both craftsmen find themselves having to deal with a resultant angle. This situation is caused by the fact that both the hip rafter and the chair leg are not simply angled in one direction. They’re both angled in two directions. To make matters worse, the reference surfaces may, or may not be visible. What’s a body to do? Well, imagine that the gods of carpentry have dropped one great big plumb line from the cosmos and use that as your reference. I’m serious here.
Several years ago, I was reading a post on Peter Galbert’s blog, Chairnotes. In it he mentioned a method of using simple geometry as a means of finding resultant and sighting angles when the rake and splay are known. Anyone who ever bored holes for “plank” chair legs is familiar with the problem. I, along with a great many other chair makers, are fortunate enough to own a copy of Drew Langsner’s book, The Chairmaker’s Workshop. The book is a gift to mankind. It is an incredible source of knowledge and worth any price you have to pay for it, if for no other reason than the rake/splay/result/sight tables in the back of the book. I was always a little worried that I might loose “the book”. Or, a good friend might want to borrow it and forget to return it. The house might burn down. I knew I could never replace the higher mathematical treasure that it contained.
Then I read Galbert’s post and a bell went off. It was like I was “Pavlov’s” dog. My salivary glands began to work. I was overwhelmed with the warm feeling of familiarity. I knew this theory! I mean, hello! Guys have been dealing with resultant angles long before Leibniz and Newton ever started being “pissy” with one another. Think pyramids! Simple geometry! This was simply a geometric method of determining the length and cut angles of a hip rafter (without having to hand it up the ladder to the youngest member of the crew). This was stuff my Grandfather expected me to remember. If Hiram Abiff and Pythagoras could have met for a beer, this is the kind of stuff they’d be talking about.
Right now we’re talking chairs, stools – we’ll leave hip rafters for another time (but the principle is the same). First consider the plumb line. For this exercise the plumb line length will be equal to the height of the seat from the floor. (If we were building a house, this would be the “rise”.) Remember to take into account seat slope, if there is any. Also, for the purpose of calculation, it’s probably a good idea to work from center lines, remembering the varying thickness of things like seat planks. And don’t forget, if the piece you’re working on has varying rakes and splays for front and rear legs, you’ll have to make two separate calculations.
First, lay two perpendicular lines:
Then swing an arc equal to the length of the plumb line,through, at least, three lines:
Now, lay lines to form two right triangles, representing the rake and splay. This can be done in degrees or any unit of measurement you care to use. It may be easier to visualize the positioning of the triangles by imagining that you’re standing in front of and looking down at the chair seat (Remember the chair is “stage” right, someone else is sitting in it, not you.) The rake is coming back toward you and the splay is running to the side:
Next, draw two 90 degree lines from the points at which the hypotenuses intersect the perpendicular lines:
Now draw a diagonal line from the center point to the corner of the rectangle. This diagonal line is your sight line. Next, raise a line at 90 degrees from the diagonal at the center point and construct the third triangle. This is your resultant angle.
Unless your rake and splay angles are equal, the length of the hypotenuses will vary, with the resultant angle having the longest. No more guessing at leg length (assuming your taper is correct or your shoulder is placed properly).
I’m still going to take very good care of Brother Langsner’s book. But, should some blackguard, n’er-do-well, or thief in the night, separate me from that magnificent tome, I could probably “limp along” for a while. Thank you Messers Pythagoras and Euclid. QED